Let $ (c_n)_{n\geq 0} $ be a sequence of positive reals such that $ \dfrac{1}{m}\sum_{k=0}^{m-1}c_{k}\sim\prod_{k=0}^{m-1} c_{k}$ as $ m $ tends to infinity. Call such a sequence a "corridor sequence" (as intuitively each of the terms should be "close to $ 1 $).

Which upper bound can be given for the quantity $ \sup_{k\leq m}\{c_{k}\}-\inf_{k\leq m}\{c_{k}\} $ provided the set of values of the corridor sequence is dense in some interval containing $ 1 $ ?